MA725 Differential Geometry

Research

My research interests are in
differential geometry in finite and infinite dimensions,
particularly with applications to/from
mathematical physics.
Almost all this work
uses Laplacian-type operators
sooner or later.
My current work focuses on characteristic classes for infinite dimensional
bundles, with collaborations with Andres Larrain-Hubach, Yoshi Maeda, Sylvie
Paycha, Simon Scott, and Fabian Torres-Ardila.
Older work
includes the functional/zeta
determinant of Laplacians, which is a key element of quantum field
theory (or non-theory), and (with K. D. Elworthy and
Xue-Mei Li)
applications of Brownian motion to differential geometry. This has
given a series of results of the type:
topological condition A on a compact manifold implies that metrics of
type B
cannot exist on the manifold. In particular, these theorems extend
the classical Bochner and Myers type theorems.
Heat
operators associated to Laplacians figure heavily in this work; after
all, Brownian motion is supposed to model heat flow as an example of
infinite dimensional Riemannian geometry. More recently, in a series of papers
with Andres Larrain-Hubach, Yoshiaki Maeda, Sylvie Paycha, Simon Scott
and Fabian Torres-Ardila, we've studied
primary and secondary characteristic classes on infinite dimensional manifolds
such as loop spaces; here the Laplacians enter in the curvature
of connections on these manifolds

Other work: Yoshiaki Maeda, Philippe Tondeur and I have worked on the
geometry of the gauge orbits in the space of connections, and on the
geometry of the orbits of the diffeomorphism group
in the space of
metrics on a manifold.
Mihail Fromosu and I have studied Mathai-Quillen forms,
which have formal applications in QFT and rigorous applications in
differential geometry. There
are
also preprints on length spaces (with Deane Yang), on
quantum cohomology (with Mihaela Vajiac), on Lax pairs and Feynman
diagrams (with Gabriel Baditoiu), and on applications of differential topology
to analysis of networks (with Cedric Geneset, Prakash Balanchandran, and Eric
Kolaczyk).

My book, "The Laplacian on a Riemannian
Manifold," now in its second (corrected) printing, is also available.
This book is aimed at graduate students who have had a
basic course in manifolds through integration of forms. The goal is
to get students to appreciate current areas of research in global geometry.
The book
covers Hodge theory, basics of differential geometry, heat flow on
functions and forms, the heat equation/supersymmetric proof of the
Chern-Gauss-Bonnet theorem, an overview of the Atiyah-Singer Index
Theorem, the zeta function for Laplacians and analytic torsion.
There are lots of exercises. You
can preview the
introduction. The price is $128.00 for hardcover (ISBN 0 521 46300 9)
and $51.00 for paperback (ISBN 0 521 46831 0).
You can order copies (no
upper limit) from the publisher, Cambridge
University Press. Alternatively, you can view the book
here. Feel free to print it out, but consider
making a donation to a good cause in lieu of buying the text. I would like to
thank Cambridge University Press for allowing me to make the text available
online, in contrast to the attitude of other math text publishers.

For an antiquated streaming video of a lecture at MSRI in 1998, click
here and select the "start 56kps video" option in the
top left corner. Good luck.